Final answer:
The vertical component of the electric field at point P due to a uniformly charged rod is found by integrating the vertical electric field contributions of infinitesimal charge elements along the rod, taking into account that the horizontal components cancel out.
Step-by-step explanation:
To calculate the vertical component of the electric field (Ey) at point P due to a uniformly charged rod, we consider each infinitesimal charge element dq along the length L of the rod. The contribution of electric field dE due to an infinitesimal charge dq at any point on the rod to point P has a vertical component dEy. To find the total electric field Ey at point P, we integrate these infinitesimal vertical components across the entire length of the rod.
The total charge Q is uniformly distributed along the rod, so the charge per unit length is λ = Q / L. Using Coulomb's law, the electric field due to a point charge is E = k*q / r^2, where k is Coulomb's constant and r is the distance from the charge to the point of interest. The total electric field at point P can then be found by integrating the contributions from each point on the rod. Note that only the component perpendicular to the rod contributes to Ey because the horizontal components cancel out due to symmetry.